Question: The lifespans of meerkats in a particular zoo are normally distributed. The average meerkat lives $10.4$ years; the standard deviation is $1.9$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a meerkat living between $12.3$ and $14.2$ years.
Answer: $10.4$ $8.5$ $12.3$ $6.6$ $14.2$ $4.7$ $16.1$ $95\%$ $68\%$ $13.5\%$ $13.5\%$ We know the lifespans are normally distributed with an average lifespan of $10.4$ years. We know the standard deviation is $1.9$ years, so one standard deviation below the mean is $8.5$ years and one standard deviation above the mean is $12.3$ years. Two standard deviations below the mean is $6.6$ years and two standard deviations above the mean is $14.2$ years. Three standard deviations below the mean is $4.7$ years and three standard deviations above the mean is $16.1$ years. We are interested in the probability of a meerkat living between $12.3$ and $14.2$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $95\%$ of the meerkats will have lifespans within 2 standard deviations of the average lifespan. It also tells us that $68\%$ of the meerkats will have lifespans within 1 standard deviation of the mean. The probability of a particular meerkat living between $12.3$ and $14.2$ years is $\color{orange}{13.5\%}$.